Eval calls apply,
which just calls eval again!
When does it all end?
In this project, you will develop an interpreter for a subset of the Scheme language. As you proceed, think about the issues that arise in the design of a programming language; many quirks of languages are the byproduct of implementation decisions in interpreters and compilers.
You will also implement some small programs in Scheme, including the
count_change function that we studied in lecture. Scheme is a simple but
powerful functional language. You should find that much of what you have learned about
Python transfers cleanly to Scheme as well as to other programming languages. To learn
more about Scheme, you can read the original Structure
and Interpretation of Computer Programs online for free. Examples from
chapters 1 and 2 are included as test cases for this project. Language
features from Chapters 3, 4, and 5 are not part of this project, but of
course you are welcome to extend your interpreter to implement more of the
language. Since we only include a subset of the language, your interpreter
will not match exactly the behavior of other interpreters such as STk.
The project concludes with an open-ended graphics contest that challenges you to produce recursive images in only a few lines of Scheme. As an example of what you might create, the picture above abstractly depicts all the ways of making change for $0.50 using U.S. currency. All flowers appear at the end of a branch with length 50. Small angles in a branch indicate an additional coin, while large angles indicate a new currency denomination. In the contest, you too will have the chance to unleash your inner recursive artist.
This project includes several files, but all of your changes will be made to the
tests.scm. You can download all of the
project code as a zip archive.
||The Scheme evaluator|
||The Scheme parser|
||A collection of test cases written in Scheme that are designed to test your Scheme interpreter. You will implement some Scheme procedures yourself.|
||A tokenizer for scheme|
||Primitive Scheme procedures|
||A testing framework for Scheme|
||Utility functions for 61A|
This is a two-part project. As in the previous project, you'll work in a team of two people, person A and person B. All questions are labeled sequentially, but some are designated for certain people by a prefix of their letter (A or B). Both partners should understand the solutions to all questions.
In the first part, you will develop the interpreter in stages:
In the second part, you will implement Scheme procedures that are similar to some exercises that you previously completed in Python.
There are 27 possible correctness points and 3 composition points. The composition score in this project will evaluate the clarity of your code and your ability to write tests that verify the behavior of your interpreter.
Before you begin working on the project, review what you have learned in lecture about the Scheme language in Chapter 2.5 of the course lecture notes.
Read-Eval-Print. Run interactively, the interpreter reads Scheme
expressions, evaluates them, and prints the results. The interpreter uses
scm> as the prompt.
scm> 2 2 scm> (((lambda (f) (lambda (x) (f f x))) (lambda (f k) (if (zero? k) 1 (* k (f f (- k 1)))))) 5) 120
The starter code for your Scheme interpreter in
can successfully evaluate the first expression above, since it consists of a
single literal numeral. The second (a computation of 5 factorial) will not
work just yet.
load function differs from standard Scheme
in that we use a symbol for the file name. For example,
scm> (load 'tests)
Symbols. Unlike some implementations of Scheme, in this project numbers and boolean values cannot be used as symbols. Symbols may not be capitalized.
Turtle Graphics. In addition to standard Scheme procedures, we
include procedure calls to the Python
turtle package. Read its
turtle Python module may not
be installed by default on your personal computer. However, the
turtle module is installed on the instructional machines.
So, if you wish to create turtle graphics for this project (i.e. for
the contest), then you'll either need to setup
your personal computer, or test on your class account.
The tests for this project are largely taken from the Scheme
textbook that 61A used for many years. Examples from relevant
chapters (and a few more examples to test various corner cases)
You can also compare the output of your interpreter to the expected output
tests.scm file contains Scheme
expressions interspersed with comments in the form:
(+ 1 2) ; expect 3 (/ 1 0) ; expect Error
scheme_test.py will evaluate
(+ 1 2)
using your code in
scheme.py, then output a test failure
3 is not returned. The second example tests for an error
(but not the specific error message). The
collects these expected outputs and compares them with the actual output from
the program, counting and reporting mismatches.
Only a small subset of tests are designated to run by default because
tests.scm contains an
(exit) call near the
beginning, which halts testing. As you complete more of the project, you
should move or remove this call. Note that your interpreter doesn't know
how to exit until Problems 3 and 4 are completed.
Important: As you proceed in the project, add new tests to the top
tests.scm to verify the behavior of your implementation.
Finally, as always, you can run the doctests for the project using:
python3 -m doctest scheme.py scheme_reader.py
Don't forget to use the
trace decorator from the
ucb module to follow the path of execution in your
As you develop your Scheme interpreter, you may find that Python raises
various uncaught exceptions when evaluating Scheme expressions. As a result,
your Scheme interpreter will crash. Some of these may be the results of bugs
in your program, and some may be useful indications of errors in user
programs. The former should be fixed (of course!) and the latter should be
caught and changed into
SchemeError exceptions, which are caught
and printed as error messages by the Scheme read-eval-print loop we've
written for you. Python exceptions that "leak out" to the user in raw form
are errors in your interpreter (tracebacks are for implementors, not
To run your Scheme interpreter in an interactive mode, type:
python3 scheme.pyAlternately, you can tell your Scheme interpreter to evaluate the lines of an input file by passing the file name as an argument to
python3 scheme.py tests.scmCurrently, your Scheme interpreter can handle a few simple expressions, such as:
scm> 1 1 scm> 42 42 scm> #t TrueTo exit the Scheme interpreter, issue either
Ctrl-dor evaluate the
buffer.py) instance that returns
valid Scheme tokens on invocations of
pop methods. It returns the next full Scheme expression in the
src buffer, using an internal representation as follows:
|Scheme Data Type||Our Internal Representation|
|Numbers|| Python's built-in
|Symbols|| Python's built-in
| Booleans (
|| Python's built-in
Problem 1 (1 pt). Complete the
scheme_read function in
scheme_reader.py by adding support for quotation.
srcis the string
"nil", return the
'bagel), then treat the quoted symbol as the special form
"(", return the result of
Problem 2 (2 pt). Complete the
read_tail function in
scheme_reader.py by adding support for dotted lists. A dotted
list in Scheme is not necessarily a well-formed list, but instead has an
second attribute that may be any Scheme value.
read_tail function expects to read the rest of a list or
dotted list, assuming the open parenthesis has already been popped by
Consider the case of calling
scheme_read on input "
. 3)". The
read_tail function will be called on the
1 2 . 3)", which is
1and the value of the tail "
2 . 3)", which is
2and the Scheme value
Pair(1, Pair(2, 3)).
Hint: In order to verify that only one element follows a dot, after
'.', read one additional expression and then
check to see that a closing parenthesis follows.
To verify that your solutions to Problem 1 and 2 work correctly, run the
scheme_reader.py and test your parser interactively
# python3 scheme_reader.py read> 42 42 read> '(1 2 3) (quote (1 2 3)) read> nil () read> '() (quote ()) read> (1 (2 3) (4 (5))) (1 (2 3) (4 (5))) read> (1 (9 8) . 7) (1 (9 8) . 7) read> (hi there . (cs . (student))) (hi there cs student)
All further changes to the interpreter will be made in
scheme.py. For each question, add a few tests to the top of
tests.scm to verify the behavior of your implementation.
3.7 of the course lecture notes describes the structure of the Scheme
evaluator. In the implementation given to you, the
function is complete, but few of the functions or methods it uses are
implemented. In fact, the evaluator can only evaluate self-evaluating
expressions: numbers, booleans, and
Problem 3 (2 pt). Implement
apply_primitive, which is
Primitive procedures are applied by calling a corresponding Python function
that implements the procedure.
Scheme primitive procedures are represented as instances of the
PrimitiveProcedure class, defined in
PrimitiveProcedure has two
self.use_envis a boolean flag that indicates whether or not this primitive procedure will expect the current environment to be passed in as the last argument. The environment is required, for instance, to implement the primitive
To see a list of all Scheme primitive procedures used in the project, look
scheme_primitives.py file. Any function decorated with
@primitive will be added to the globally-defined
apply_primitive function takes a
PrimitiveProcedure instance, a Scheme list of argument values,
and the current environment. Your implementation should:
True, then add the current environment
envas the last argument.
procedure.fnon those arguments (hint: use * notation).
TypeErrorexception being thrown, then raise a
The doctest for
apply_primitive should now pass. However,
your Scheme interpreter will still not be able to apply primitive
procedures, because your Scheme interpreter still doesn't know
how to look up the values for the primitive procedure symbols (such as
Problem 4 (2 pt) Implement the
lookup method of the
Frame class. It takes a symbol (Python string) and returns the
value bound to that name in the first frame of the environment in which it is
Frame represents an environment via two instance
self.bindingsis a dictionary that maps Scheme symbols (represented as Python strings) to Scheme values.
self.parentis the parent
Frameinstance. The parent of the Global Frame is
self.bindingsif it exists.
lookupthat symbol in the parent if it exists.
After you complete this problem, you should be able to evaluate primitive procedure calls, giving you the functionality of the Calculator language and more.
scm> + <scheme_primitives.PrimitiveProcedure object at 0x2742d50> scm> (+ 1 2) 3 scm> (* 3 4 (- 5 2) 1) 36 scm> (odd? 31) True
Problem A5 (1 pt). There are two missing parts in the method
do_define_form, which handles the
special forms. Implement just the first part, which binds names to values
but does not create new procedures.
scm> (define tau (* 2 3.1415926))
You should now be able to give names to values and evaluate symbols to those values.
scm> (define x 15) scm> (define y (* 2 x)) scm> y 30 scm> (+ y (* y 2) 1) 91 scm> (define x 20) scm> x 20
Problem B6 (1 pt). Implement the
function, which evaluates the
quote special form. Once you have
done so, you can evaluate quoted expressions.
scm> 'hello hello scm> '(1 . 2) (1 . 2) scm> '(1 (2 three . (4 . 5))) (1 (2 three 4 . 5)) scm> (car '(a b)) a scm> (eval (cons 'car '('(1 2)))) 1
At this point in the project, your Scheme interpreter should be be able to support the following features:
(+ (- 4 2) 5)
In our interpreter, user-defined procedures will
be represented as instances of the
instance has three instance attributes:
self.formalsis a Scheme list of the formal parameters (symbols) that name the arguments of the procedure.
self.bodyis a single Scheme expression; the body of the procedure.
self.envis the environment in which the procedure was defined.
Problem 7 (2 pt). Implement the
begin special form,
which has a list of one or more sub-expressions that are each evaluated in
order. The value of the final sub-expression is the value of the
scm> (begin (+ 2 3) (+ 5 6)) 11 scm> (begin (display 3) (newline) (+ 2 3)) 3 5
scheme_eval evaluates one of
the conditional constructs (
case), notice that it calls
on the return value of the relevant
Take care that your Scheme interpreter doesn't inadvertantly call
scheme_eval on the same value twice, or else you might
get the following invalid behavior:
scm> (begin 30 'hello) Error: unknown identifier: hello
Problem 8 (2 pt). Implement the
LambdaProcedure values by evaluating
lambda expressions. While you cannot call a user-defined
procedure yet, you can verify that you have read the procedure correctly by
evaluating a lambda expression.
scm> (lambda (x y) (+ x y)) (lambda (x y) (+ x y))In Scheme, it is legal to have function bodies with more than one expression:
STk> ((lambda (y) 42 (* y 2)) 5) 10In order to implement this feature, your
do_lambda_formshould detect when the body of a lambda expression contains multiple expressions. If so, then
do_lambda_formshould place the expressions inside of a
(begin ...)expression, and use that
beginexpression as the body:
scm> (lambda (y) 42 (* y 2)) (lambda (y) (begin 42 (* y 2)))
Problem A9 (1 pt). Currently, your Scheme interpreter is able to define user-defined procedures in the following manner:
scm> (define f (lambda (x) (* x 2)))However, we'd like to be able to use the shorthand form of defining procedures:
scm> (define (f x) (* x 2))
do_define_form function so that it correctly
handles the shorthand procedure definition form above. Make sure that it can
handle multi-expression bodies. Hint: construct a
expression and evaluate it with
Once you have completed this problem, you should find that defined procedures evaluate to lambda procedures.
scm> (define (square x) (* x x)) scm> square (lambda (x) (* x x))
Problem 10 (2 pt). Implement the
make_call_frame method of
Frame class. It should:
Problem B11 (1 pt). Implement the
function to raise an error whenever the Scheme list of formal parameters
passed to it is invalid. Raise a
SchemeError if the list of
formals is not a well-formed list of symbols or if any symbol is
Problem 12 (2 pt). Implement
scheme_apply to correctly
LambdaProcedure instances. (The case of
MuProcedures is handled later in the project). It should:
Frame, with all formal parameters bound to their argument values.
procedurein the environment represented by this new frame.
After you complete
scheme_apply, user-defined functions (and
lambda functions) should work in your Scheme interpreter. Now is an
excellent time to revisit the tests in
tests.scm and ensure that
you pass the ones that involve definition (Sections 1.1.2 and 1.1.4). You
should also add additional tests of your own to verify that your interpreter
is behaving as you expect.
The basic Scheme logical special forms are
cond. These expressions
are special because not all of their sub-expressions may be evaluated.
In Scheme, only
#f (also known as
False) is a false value. All other values are true values. You
can test whether a value is a true value or a false value using the provided
Problem A13 (1 pt). Implement
do_if_form so that
if expressions are evaluated correctly. This function should
return either the second (consequent) or third (alternative) expression of
if expression, depending on the value of the first
Note: For this project, we will only handle
expressions that contain three operands. The following
expressions should be correctly supported by your interpreter:
scm> (if (= 4 2) true false) False scm> (if (= 4 4) (* 1 2) (+ 3 4)) 2And the following expression should be rejected by your interpreter:
scm> (if (= 4 2) true) Error: too few operands in form
Problem B14 (2 pt). Implement
do_or_form so that
expressions are evaluated correctly.
The logical forms
or are short-circuiting.
and, your interpreter should evaluate each argument from
left to right, and if any argument evaluates to
False is returned. If all but the last sub-expressions evaluate
to true values, return the last sub-expression from
or evaluate each argument from left to right, amd
if any argument evaluates to a true value, then return it. If all but the
last sub-expression evaluate to false, return the last sub-expression from
scm> (and) True scm> (or) False scm> (and 4 5 6) 6 ; all operands are true values scm> (or 5 2 1) 5 ; 5 is a true value scm> (and #t #f 42 (/ 1 0)) False ; short-circuiting behavior of and scm> (or 4 #t (/ 1 0)) 4 ; short-circuiting behavior of or
Problem A15 (1 pt). Implement
do_cond_form so that it
returns the first result sub-expression corresponding to a true predicate (or
else). Your implementation should match the following examples and the
additional tests in
scm> (cond ((= 4 3) 'nope) ((= 4 4) 'hi) (else 'wait)) hi scm> (cond ((= 4 3) 'wat) ((= 4 4)) (else 'hm)) True scm> (cond ((= 4 4) 'here 42) (else 'wat 0)) 42For the last example, where the body of a
condhas multiple expressions, you might find it helpful to replace
cond-bodies with multiple expression bodies into a single
beginexpression, i.e., the following two expressions are equivalent.
(cond ((= 4 4) 'here 42)) (cond ((= 4 4) (begin 'here 42)))
A few clarifications:
elseclauses must contain at least one expression, i.e. the following
condusage is invalid in our project:
scm> (cond ((= 4 3) 5) (else))
condis satisfied, then your
None, which signals an undefined expression.
Problem A16 (2 pt). The
let special form introduces local
variables, giving them their initial values. For example,
scm> (define x 'hi) scm> (define y 'bye) scm> (let ((x 42) (y (* 5 10))) (list x y)) (42 50) scm> (list x y) (hi bye)Implement the
do_let_formmethod to have this effect and test it, by adding test cases to
tests.scm. Make sure your
letcorrectly handles multi-expression bodies:
scm> (let ((x 42)) x 1 2) 2
The let special form is equivalent to creating and then calling a lambda procedure. That is, the following two expressions are equivalent:
(let ((x 42) (y 16)) (+ x y)) ((lambda (x y) (+ x y)) 42 16)Thus, a
letform implicitly creates a new
letbindings) which extends the current environment and evaluates the body of the
letwith respect to this new
Frame. This is very much exactly like a user-defined function call. Note that, in your project code, you don't have to actually create a
LambdaProcedureand call it. Instead, you can create a new
Frame, add the necessary bindings, and evaluate the expressions of the
letbody with respect to the new
The bindings created by a
let are not able to refer back to
previously-declared bindings from the same
Problem B17 (2 pt). Implement
do_mu_form to evaluate
mu special form, a non-standard Scheme expression type. A
mu expression is similar to a
but evaluates to a
MuProcedure instance that is dynamically
MuProcedure class has been provided for you.
scheme_apply to call
MuProcedure procedures using dynamic scoping. Calling a
LambdaProcedure uses lexical scoping: the parent of the new call
frame is the environment in which the procedure was defined. Calling a
MuProcedure created by a
mu expression uses dynamic
scoping: the parent of the new call frame is the environment in which the
call expression was evaluated. As a result, a
not need to store an environment as an instance attribute. It can refer to
names in the environment from which it was called.
scm> (define f (mu (x) (+ x y))) scm> (define g (lambda (x y) (f (+ x x)))) scm> (g 3 7) 13
Your Scheme interpreter implementation is now complete. You should have
been adding tests to
tests.scm as you did each problem. These
tests will be evaluated as part of your composition score for the project.
Make sure that your project works as expected.
Not only is your Scheme interpreter itself a tree-recursive program, but it is
flexible enough to evaluate other recursive programs. Implement the following
procedures in Scheme at the bottom of
Problem 18 (2 pt). Implement the
merge procedure, which takes
two sorted list arguments and combines them into one sorted list. For example:
scm> (merge '(1 4 6) '(2 5 8)) (1 2 4 5 6 8)
Problem A19 (2 pt). Implement the
count-change procedure, which
counts all of the ways to make change for a
total amount, using coins with
various denominations (
denoms), but never uses more than
max-coins in total. The procedure definition line is provided
tests.scm, along with a list of U.S. denominations.
Problem B20 (2 pt) Implement the
which counts all the ways to partition a positive integer
total using only
pieces less than or equal to another positive integer
5 has 5 partitions using pieces up to a
3, 2 (two pieces) 3, 1, 1 (three pieces) 2, 2, 1 (three pieces) 2, 1, 1, 1 (four pieces) 1, 1, 1, 1, 1 (five pieces)
Problem 21 (2 pt). Implement the
which lists all of the ways to partition a positive integer
total into at
max-pieces pieces that are all less than or equal to a positive
max-value. Hint: Define a helper function to construct
Problem 22 (0 pt). Implement the
hax procedure that
draws the following recursive illustration when passed two arguments, a side
d and recursive depth
k. The example below
is drawn from
(hax 200 4).
To see how this illustration is constructed, consider this annotated version that gives the relative lengths of lines of the component shapes in the figure.
Problem 23 (5 pt). Complete the function
scheme.py. This alternative to
properly tail recursive. That is, the interpreter will allow an unbounded
number of active tail
calls in constant space.
Instead of recursively calling
scheme_eval for tail calls and
logical special forms, and
let, replace the current
env with different expressions and
environments. For call expressions, this change only applies to calling
Once you finish, uncomment the line
scheme_eval = scheme_optimized_eval in
Congratulations! You have finished the final project for 61A! Assuming your tests are good and you've passed them all, consider yourself a proper computer scientist!
Now, get some sleep. You've earned it!
We've added a number of primitive drawing procedures that are collectively
called "turtle graphics". The turtle represents the state of the drawing
module, which has a position, an orientation, a pen state (up or down), and a
pen color. The
tscheme_x functions in
scheme_primitives.py are the implementations of these
procedures, and show their parameters with a brief description of each.
The Python documentation of
the turtle module contains more detail.
Contest (3 pt). Create a visualization of an iterative or recursive process of your choosing, using turtle graphics. Your implementation must be written entirely in Scheme using the interpreter you have built.
Prizes will be awarded for the winning entry in each of the following categories.
Entries (code and results) will be posted online, and winners will be selected by popular vote as part of a future homework. The voting instructions will read:
Please vote for your favorite entry in this semester's 61A Recursion Exposition contest. The winner should exemplify the principles of elegance, beauty, and abstraction that are prized in the Berkeley computer science curriculum. As an academic community, we should strive to recognize and reward merit and achievement (translation: please don't just vote for your friends).
To improve your chance of success, you are welcome to include a title and descriptive haiku in the comments of your entry, which will be included in the voting.
Submission instructions will be posted shortly.